Abstract

We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian
spaces. Moreover, conditions of continuity of the extended transform and its inverse with
respect to δ and Δ convergence are discussed in some details.

1. Introduction

Optical integral transforms have been studied in several works, for example, [1–8]. However, is the Fresnel transform among all the great importance [5, 9] where for which the kernel takes the form of a complex exponential function , for some constants , and . The generalization of the Fresnel transform called the linear canonical transform was introduced in [10] and has recently attracted considerable attention in optics, see [4, 11]. One of the very well-known linear transform is the wavelet transform, see [12, 13] we have
where is named as the mother wavelet such that
and are the transform dilate and translate of the wavelet and being the complex conjugate of . The optical diffraction transform is described by the Fresnel integration in [5, 9] as follows:

The parameters are elements of more ray transfer Matrix describing optical systems, . For a details of Fresnel integrals, see [14, 15].

Note that many familiar transforms can be considered as special cases of the diffraction Fresnel transform. For example, if the parameters , and are written in the following matrix form:
then the diffraction Fresnel transform, the generalized Fresnel Transform becomes a fractional Fourier transform, see [11, 16, 17].

In the present work, we consider a combined optical transform of Fresnel and wavelet transforms, namely, the optical Fresnel-wavelet transform defined by [9]
with kernel

The parameters , and appearing in (1.5) are elements of matrix with unit determinant.

As the general single-mode squeezing operator of the generalized Fresnel transform is in wave optics, further its applications are having a faithful representation in the optical Fresnel-wavelet transform, see [9]. Therefore the combined optical Fresnel-wavelet transform can be more conveniently studied by the general single-mode squeezed operation.

However, our discussion is somewhat different and making more interesting. Since the theory of the optical Fresnel-wavelet transform of generalized functions has not been reported in the literature. Thus, we extend the optical Fresnel-wavelet transform to a specific space of generalized functions, namely, known as Boehmian space. In Section 2, we observe that the kernel function of the Fresnel-wavelet transform is a smooth function, and therefore the optical Fresnel-wavelet transform is defined as an adjoint operator in the space of distributions. In a concrete way, Section 3 builds an appropriate space of Boehmians, whereas Section 4 constructs a new space of all images of Boehmians from Section 3. In Section 5, we define the optical Fresnel-wavelet transform of a Boehmian and study some of its general properties.

2. Optical Fresnel-Wavelet Transforms of Distributions

Let be the space of all test functions of arbitrary support and be its dual of distributions of bounded support, see, for example [12, 18–20]. Then, is a complete multinormed space with the set of norms as follows:
where run through compact subsets of and . It is clear that the kernel function of the optical Fresnel-wavelet transform
for each , is an element of . This describes the distributional optical Fresnel-wavelet transform of bounded support as an adjoint operator as follows:
For convenience we sometimes write instead of . Moreover, from (2.3), we observe that is an analytic function satisfying the expression as follows:

Further, is well defined since has its usual meaning where denoted by to be the usual convolution product [18, 20, 21]. Then we have the following lemma.

Lemma 2.1. Let and , be their respective optical Fresnel-wavelet transforms and for all and , then

where .

Proof. Let , and , and then
Hence, using properties of distributions and simple calculations we get
The above theorem is known as the convolution theorem of the Fresnel-wavelet transform.

Let be the dirac delta function. Then the Fresnel-wavelet transform of is described as follows:

3. The Boehmian Space

In this section, we assume that the reader is acquainted with the general construction of Boehmian spaces [6, 22–27]. Let be the Schwartz space of test functions of bounded support see [12, 20, 28]. The operation between a distribution and a test function is defined by
where and

A sequence of functions in is said to be a delta sequence if it satisfies Conditions (3.3)–(3.5). Consider

The set of all such sequences is denoted by . To see the extension to certain integral transform, see [29–31].

Lemma 3.1. Given and then
for each .

Proof. We prove the lemma by induction on . Let and with , and then
Hence (3.7) reduces to

Next, assume that the lemma satisfies for th derivatives, then certainly we get
by (3.7). Hence the lemmais as follows.

Lemma 3.2. Let and , and then .

Proof. Let be a compact subset of . Then using Lemma 3.1 we get
The inequality (3.10) can be explicitly expressed as
where is the norm in the topology equipped with . Hence, the lemma follows from (3.11). This completes the proof.

Lemma 3.3. Let and , and then .

Proof. In view of Lemma 3.2 we get
Therefore, the righthand side of (3.1) is meaningful. To show that we are requested to show that is continuous and linear. To establish continuity, let in , then from (3.11) we get

Hence we have
as . Linearity condition is obvious. Hence the lemma is completely proved.

Lemma 3.4. Let and be given, and then

Proof. It is a straightforward consequence of definitions and change of variables.

Lemma 3.5. Let and , and then

Proof. Using (3.1) and Lemma 3.4. we get
Hence the lemma is as follows.

Lemma 3.6. Let and , and then one has (1)(2).

Proof. It is a straightforward result of definitions.

Lemma 3.7. Let in and be given then
as .

Proof. By virtue of Lemma 3.2, . Hence, using (3.1) we get
Allowing completes the proof of the lemma.

Lemma 3.8. Let and , and then .

Proof. Considering a compact subset of and a sequence such that for each , we show that as in the sense of . By Lemma 3.1 we have
and by applying (3.4) we get
Then the mean value theorem implies that
for some . Let then considering supremum over all with the fact that yields
Now allowing in (3.23) yields . Hence we have established that
On using (3.24) can be observed as
This implies that as . The proof is therefore completed.

Finally, by virtue of the above sequence of results (Lemma 3.1–3.8), our desired Boehmian space is well defined.

4. The Boehmian Space

In this section we construct the space of all Fresnel-wavelet transforms of Boehmians from the space as follows.

Let be the space of all analytic functions which are Fresnel-wavelet transforms of distributions in . Then, we define convergence as follows. We say in if and only if there are such that in , where and . Let and be given, and then define

Theorem 4.1. Let and , and then , where .

Proof. By the aid of (2.3) we write
By using (3.1) we get
That is . Employing (4.1) yields , where . This proves the theorem.

Lemma 4.2. Let and , and then .

Proof. Let be such that , and that then Theorem 4.1 implies
Thus by Lemma 3.3. Hence the lemma.

Lemma 4.3. Let and , and then .

Proof. Theorem 4.1 implies that . This completes the proof of the lemma.

Lemma 4.4. Let and , then (1),
(2).

Proof. It is obvious.

Lemma 4.5.
(i) Let as and , and then as in .
(ii) Let as and , and then as in .

Proof.
(i) Let then and for some . Hence, using Theorem 4.1, we have
By Lemma 3.7 we get
Once again Theorem 4.1 implies that
Thus we ahve
This proves part (i) of the Lemma.
(ii) can be proved similarly by using Lemma 3.8 and Theorem 4.1. The space is therefore established.

The sum of two Boehmians and multiplication by a scalar in is defined in a natural way as follows:

The operation and the differentiation are defined by

5. Optical Fresnel-Wavelet Transforms of Boehmians

In view of the analysis obtained in Sections 4 and 5 and Theorem 4.1 we are led to state the following definition.

Proof. Assume that , and then using (5.1) and the concept of quotients we get , where and . Therefore, Theorem 4.1 implies that . Properties of imply that . Therefore, . To establish is surjective, and let . Then for every . Hence are such that and . Theorem 4.1 implies that . Hence is such that . This completes the proof of the lemma.

Now, Let , and then we define the inverse optical Fresnel-wavelet transform of as
where .

Proof. First of all, we show that and are continuous with respect to convergence. Let in as , and then we show that as . By virtue of [25] we can find and in such that
such that as for every . Employing the continuity condition of the optical Fresnel-wavelet transform implies that as in the space . Thus, as in . To prove the second part, let in as . Then, once again, by [4], and for some and as . Hence in as . Or, as . Using Definition 5.1 we get as . Now, we establish continuity of and with respect to convergence. Let in as . Then, there exist and such that and as . Employing Definition 5.1 we get
Hence, from Lemma 4.2 we have as in . Therefore consider
Hence, as . Finally, let in as , and then we find such that and as for some and . Now, using Definition 5.1, we obtain that
Lemma 5.5 implies that
Thus we have
From this we find that as in . This completes the proof of the theorem.

Acknowledgment

The authors would like to express their sincere thanks and gratitude to the reviewer(s) for their valuable comments and suggestions for the improvement of this paper.

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